- FindStat

Identifier

Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000378: Integer partitions ⟶ ℤ

Values

[1] => [1] => [] => 0

[1,1] => [1,1] => [1] => 1

[2] => [2] => [] => 0

[1,1,1] => [1,1,1] => [1,1] => 1

[1,2] => [2,1] => [1] => 1

[2,1] => [2,1] => [1] => 1

[3] => [3] => [] => 0

[1,1,1,1] => [1,1,1,1] => [1,1,1] => 1

[1,1,2] => [2,1,1] => [1,1] => 1

[1,2,1] => [2,1,1] => [1,1] => 1

[1,3] => [3,1] => [1] => 1

[2,1,1] => [2,1,1] => [1,1] => 1

[2,2] => [2,2] => [2] => 2

[3,1] => [3,1] => [1] => 1

[4] => [4] => [] => 0

[1,1,1,1,1] => [1,1,1,1,1] => [1,1,1,1] => 1

[1,1,1,2] => [2,1,1,1] => [1,1,1] => 1

[1,1,2,1] => [2,1,1,1] => [1,1,1] => 1

[1,1,3] => [3,1,1] => [1,1] => 1

[1,2,1,1] => [2,1,1,1] => [1,1,1] => 1

[1,2,2] => [2,2,1] => [2,1] => 3

[1,3,1] => [3,1,1] => [1,1] => 1

[1,4] => [4,1] => [1] => 1

[2,1,1,1] => [2,1,1,1] => [1,1,1] => 1

[2,1,2] => [2,2,1] => [2,1] => 3

[2,2,1] => [2,2,1] => [2,1] => 3

[2,3] => [3,2] => [2] => 2

[3,1,1] => [3,1,1] => [1,1] => 1

[3,2] => [3,2] => [2] => 2

[4,1] => [4,1] => [1] => 1

[5] => [5] => [] => 0

[1,1,1,1,1,1] => [1,1,1,1,1,1] => [1,1,1,1,1] => 1

[1,1,1,1,2] => [2,1,1,1,1] => [1,1,1,1] => 1

[1,1,1,2,1] => [2,1,1,1,1] => [1,1,1,1] => 1

[1,1,1,3] => [3,1,1,1] => [1,1,1] => 1

[1,1,2,1,1] => [2,1,1,1,1] => [1,1,1,1] => 1

[1,1,2,2] => [2,2,1,1] => [2,1,1] => 2

[1,1,3,1] => [3,1,1,1] => [1,1,1] => 1

[1,1,4] => [4,1,1] => [1,1] => 1

[1,2,1,1,1] => [2,1,1,1,1] => [1,1,1,1] => 1

[1,2,1,2] => [2,2,1,1] => [2,1,1] => 2

[1,2,2,1] => [2,2,1,1] => [2,1,1] => 2

[1,2,3] => [3,2,1] => [2,1] => 3

[1,3,1,1] => [3,1,1,1] => [1,1,1] => 1

[1,3,2] => [3,2,1] => [2,1] => 3

[1,4,1] => [4,1,1] => [1,1] => 1

[1,5] => [5,1] => [1] => 1

[2,1,1,1,1] => [2,1,1,1,1] => [1,1,1,1] => 1

[2,1,1,2] => [2,2,1,1] => [2,1,1] => 2

[2,1,2,1] => [2,2,1,1] => [2,1,1] => 2

[2,1,3] => [3,2,1] => [2,1] => 3

[2,2,1,1] => [2,2,1,1] => [2,1,1] => 2

[2,2,2] => [2,2,2] => [2,2] => 3

[2,3,1] => [3,2,1] => [2,1] => 3

[2,4] => [4,2] => [2] => 2

[3,1,1,1] => [3,1,1,1] => [1,1,1] => 1

[3,1,2] => [3,2,1] => [2,1] => 3

[3,2,1] => [3,2,1] => [2,1] => 3

[3,3] => [3,3] => [3] => 2

[4,1,1] => [4,1,1] => [1,1] => 1

[4,2] => [4,2] => [2] => 2

[5,1] => [5,1] => [1] => 1

[6] => [6] => [] => 0

[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1

[1,1,1,1,1,2] => [2,1,1,1,1,1] => [1,1,1,1,1] => 1

[1,1,1,1,2,1] => [2,1,1,1,1,1] => [1,1,1,1,1] => 1

[1,1,1,1,3] => [3,1,1,1,1] => [1,1,1,1] => 1

[1,1,1,2,1,1] => [2,1,1,1,1,1] => [1,1,1,1,1] => 1

[1,1,1,2,2] => [2,2,1,1,1] => [2,1,1,1] => 2

[1,1,1,3,1] => [3,1,1,1,1] => [1,1,1,1] => 1

[1,1,1,4] => [4,1,1,1] => [1,1,1] => 1

[1,1,2,1,1,1] => [2,1,1,1,1,1] => [1,1,1,1,1] => 1

[1,1,2,1,2] => [2,2,1,1,1] => [2,1,1,1] => 2

[1,1,2,2,1] => [2,2,1,1,1] => [2,1,1,1] => 2

[1,1,2,3] => [3,2,1,1] => [2,1,1] => 2

[1,1,3,1,1] => [3,1,1,1,1] => [1,1,1,1] => 1

[1,1,3,2] => [3,2,1,1] => [2,1,1] => 2

[1,1,4,1] => [4,1,1,1] => [1,1,1] => 1

[1,1,5] => [5,1,1] => [1,1] => 1

[1,2,1,1,1,1] => [2,1,1,1,1,1] => [1,1,1,1,1] => 1

[1,2,1,1,2] => [2,2,1,1,1] => [2,1,1,1] => 2

[1,2,1,2,1] => [2,2,1,1,1] => [2,1,1,1] => 2

[1,2,1,3] => [3,2,1,1] => [2,1,1] => 2

[1,2,2,1,1] => [2,2,1,1,1] => [2,1,1,1] => 2

[1,2,2,2] => [2,2,2,1] => [2,2,1] => 3

[1,2,3,1] => [3,2,1,1] => [2,1,1] => 2

[1,2,4] => [4,2,1] => [2,1] => 3

[1,3,1,1,1] => [3,1,1,1,1] => [1,1,1,1] => 1

[1,3,1,2] => [3,2,1,1] => [2,1,1] => 2

[1,3,2,1] => [3,2,1,1] => [2,1,1] => 2

[1,3,3] => [3,3,1] => [3,1] => 4

[1,4,1,1] => [4,1,1,1] => [1,1,1] => 1

[1,4,2] => [4,2,1] => [2,1] => 3

[1,5,1] => [5,1,1] => [1,1] => 1

[1,6] => [6,1] => [1] => 1

[2,1,1,1,1,1] => [2,1,1,1,1,1] => [1,1,1,1,1] => 1

[2,1,1,1,2] => [2,2,1,1,1] => [2,1,1,1] => 2

[2,1,1,2,1] => [2,2,1,1,1] => [2,1,1,1] => 2

[2,1,1,3] => [3,2,1,1] => [2,1,1] => 2

[2,1,2,1,1] => [2,2,1,1,1] => [2,1,1,1] => 2

[2,1,2,2] => [2,2,2,1] => [2,2,1] => 3

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Description

The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].

Map

to partition

Description

Sends a composition to the partition obtained by sorting the entries.

Map

first row removal

Description

Removes the first entry of an integer partition